Error Estimates of Finite Element Method for the Incompressible Ferrohydrodynamics Equations

Abstract

In this paper, we consider the Shliomis ferrofluid model and study its numerical approximation. We investigate a first-order energy-stable fully discrete finite element scheme for solving the simplified ferrohydrodynamics (SFHD) equations. First, we establish the well-posedness and some regularity results for the solution of the SFHD model. Next we study the Euler semi-implicit time-discrete scheme for the SFHD systems and derive the ${\varvec{L}}^2 \text -{\varvec{H}}^1$ error estimates for the time-discrete solution. Moreover, certain regularity results for the time-discrete solution are proved rigorously. With the help of these regularity results, we prove the unconditional ${\varvec{L}}^2 \text -{\varvec{H}}^1$ error estimates for the finite element solution of the SFHD model. Finally, some three-dimensional numerical examples are carried out to demonstrate both the accuracy and efficiency of the fully discrete finite element scheme.

Keywords

Shliomis model / Ferrofluids / Euler semi-implicit scheme / Mixed finite element methods / Error estimates / Unconditional convergence / 76M10 / 65N12 / 65M15 / 65M60 / 35Q35 / 35Q61

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